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Important Properties of Quadratic Equations and Functions
Definiton: A quadratic polynomial in one variable is a polynomial which may
be written in the form ax2 + bx + c where a, b, and c are real numbers and a is not 0.
Definiton: A quadratic equation in one variable is an equation which may be
written in the form ax2 + bx + c = 0 where a, b, and c are real numbers and a is not 0.
Quadratic Formula: Solutions to a quadratic equation in a single variable may be found
with the Quadratic Formula:
b  b2 4ac
x
2a
Note if b2 –4ac < 0 then
b 2 4ac is a complex number
and if b2 –4ac > 0 then
b 2 4ac is a real number
and if b2 –4ac = 0 then
b 2 4ac = 0
The Quadratic Foumula will always produce all solutions to a quadratic equation. Those
solutions may be real numbers or they may be complex numbers.
Definiton: The expression b2 –4ac is called the discriminant of the quadratic
polynomial ax2 + bx + c. We will also refer to it as the discriminant of the corresponding
quadratic equation, or the discriminant of the corresponding quadratic function.

If the discriminant of a quadratic equation in one variable is positive, the
quadratic equation has two real solutions.
o They represent two x-intercepts of the graph of the corresponding
quadratic equation in two variables.
o They also represent two x-intercepts of the graph of the corresponding
quadratic function.

If the discriminant of a quadratic equation in one variable is zero, the quadratic
equation has one real solutions.
o It represents the single x-intercept of the graph of the corresponding
quadratic equation in two variables.
o It also represents the single x-intercept of the graph of the corresponding
quadratic function.

If the discriminant of a quadratic equation in one variable is negative, the
quadratic equations has two complex solutions.
o They are conjugates of one another.
o Since only Real Numbers are represented on the real number line, these
complex solutions cannot represent x-intercepts of the graph of either the
corresponding quadratic equation in two variables or the corresponding
quadratic function.
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Definiton: A quadratic equation in two variables is an equation which may be
written in the form y = ax2 + bx + c where a, b, and c are real numbers and a is not 0.
The graph of a quadratic equation in two variables is a parabola which opens up if a > 0
and opens down if a < 0.
The y-intercept of a quadratic equation in two variables is (0, c).
The x-intercepts of the graph of a quadratic equation in two variables are found by
solving the corresponding quadratic equation in one variable.
Definiton: The vertex of a parabola which opens up is the point on the graph with the
smallest second coordinate. The vertex of a parabola which opens down is the point on
the graph with the largest second coordinate.
The vertex of the graph of a quadratic equation in two variables has first coordinate
b
.
2a
If the discriminant is positive, the graph has two x-intercepts.
If the discriminant is zero, the graph has one x-intercept and it is the vertex.
If the discriminant is negative, the graph has no x-intercepts.
It is entirely above the x-axis if a > 0 (it opens up)
It is entirely below the x-axis if a < 0 (it opens down)
Definiton: A quadratic function is a function whose rule may be written in the
form f(x) = ax2 + bx + c where a, b, and c are real numbers and a is not 0.
The graph of a quadratic function is a parabola which opens up if a > 0 and opens down if
a < 0.
The y-intercept of a quadratic function in two variables is (0, c).
Definiton: A zero of a function f is a domain element k such that f(k) = 0.
The zeros of a function are found by solving the equation resulting from f(x) = 0. In the
case of a quadratic function, the zeros are therefore found by solving the corresponding
quadratic equation in one variable. This is always possible with the quadratic formula.
The x-intercepts of the graph of a quadratic function are the real zeros of the function.
b b 

The vertex of the graph of a quadratic function is  , f  
.
2a 2a 

If the discriminant is positive, the graph has two x-intercepts.
If the discriminant is zero, the graph has one x-intercept and it is the vertex.
If the discriminant is negative, the graph has no x-intercepts.
It is entirely above the x-axis if a > 0 (it opens up)
It is entirely below the x-axis if a < 0 (it opens down)
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